Apparatus, method and system for determining credit derivative indices and estimating credit derivative credit curves, and a credit calculator for  valuing credit derivatives based on the credit curves

ABSTRACT

An apparatus, method and system for determining an estimate of at least one numerical attribute of at least one entity of a population when the population is changing and there are a limited number of observations on the attribute for the entities, in which a conditional index is determined to track how a value of the attribute changes from one time to another for an entity that is a member of the population at both times, and an unconditional index is determined representing an average level of the attribute for the entities of the population.

FIELD OF THE INVENTION

The present invention relates to an apparatus, method and system fordetermining credit derivative indices and estimating credit derivativecredit curves, and a credit calculator for valuing credit derivativesbased on the credit curves.

BACKGROUND INFORMATION

In the financial markets, there are various types of credit derivativefinancial instruments. A credit default swap (CDS) is one type of creditderivative. A CDS allows credit risks to be traded and managed in amanner similar to that of market risks. A CDS is a contract thatprovides insurance against default by a particular company. With a CDS,a seller receives a fee in exchange for making a contingent payment ifthere is a Credit Event (default) of the Reference Entity (the company).The Credit Event may be a bankruptcy, an insolvency, a receivership, amaterial adverse restructuring of debt, or a failure to meet paymentobligations when due. The CDS buyer has the right to sell the ReferenceObligation, at the Reference Obligation's par value, when the CreditEvent occurs. The total par value of the Reference Obligation (bond)that may be sold is the notional principal of the CDS. The contingentpayment may be in cash or may involve the physical delivery of theReference Obligation. The “spread” of the CDS is the total of thepayments per year, as a percent of the notional principal. The spreadmay be indicated in basis points, where 100 basis points correspond toone percent.

In particular, the buyer of the CDS makes periodic payments to theseller until the end of the life of the CDS or until a Credit Eventoccurs. A Credit Event usually requires a final accrual payment by thebuyer. The credit default swap is then settled by providing physicaldelivery or cash. If the credit default swap terms require physicaldelivery, the swap buyer delivers the bonds to the seller in exchangefor their par value. If there is a cash settlement, the “calculationagent” polls dealers to determine the mid-market price (Q) of thereference obligation a specified number of days after the Credit Event.The cash settlement is then (100−Q) percent of the notional principal.

Thus, with a CDS, a buyer gains credit protection on the ReferenceEntity and a seller assumes the default risk of the Reference Entity.

On any given trading day, there may or may not be an actual value quotedfor the spread of a CDS having a particular term and/or for a particularcompany. If there is no quote for such a CDS spread, then the CDS spread(for a particular company and/or having a particular term) must beestimated.

The CDS spread (for a particular company and/or having a particularterm) may be part of a rating category associated with a particularcredit rating or a particular industry sector, such as, for example,aerospace, automotive, steel, etc. Spreads for companies in a particularrating category are not the same, but have a tendency to move together.The companies in a particular rating category change through time. Basedon trading information, the CDS spreads have been commonly quoted forinstruments that have maturities of approximately five years.

SUMMARY OF THE INVENTION

An exemplary embodiment and/or exemplary method of the present inventionconcerns the determining of at least one estimated attribute of at leastone entity of a population of interest, in which the entities in thepopulation of interest change through time, in which for any entity inthe population, the at least one estimated attribute changes throughtime and there is a tendency for the attributes associated withdifferent entities in the population to move together, and in which on aparticular day, there are observations on the at least one attribute forsome entities.

The exemplary embodiment and/or exemplary method may include determininga conditional index for the attribute. This conditional index tracks howthe value of the attribute for an entity in the population may beexpected to change from one time to another on the condition that theentity is a member of the population at both times.

In particular, the exemplary embodiment and/or exemplary method isdirected to determining an estimate of at least one numerical attributeof at least one entity of a population of entities, in which thepopulation changes and there are a limited number of observations on theat least one numerical attribute for the at least one entity, whichincludes: determining a conditional index to track how a value of the atleast one numerical attribute changes from one time to another for anentity, in which the entity is a member of the population at both times;and determining an unconditional index representing an average level ofthe attribute for the entities that are in the population at aparticular time.

In the exemplary embodiment and/or exemplary method, the determining ofthe conditional index for the particular time may be performed bycalculating a maximum likelihood estimator for the conditional index forthe particular time.

In the exemplary embodiment and/or exemplary method, the relationshipbetween the value of the attribute and the conditional index may bedefined by x_(ij)=I_(i)+a_(j)+e_(ij), where x_(ij) is the value of theattribute for the jth entity of the population on day i, I_(i) is thelevel of the conditional index on day i, a_(j) is a constant associatedwith the jth entity of the population, and the e_(ij) have independentidentically distributed distributions.

The exemplary embodiment and/or exemplary method may further includeestimating the value of the attribute for at least one entity in thepopulation, when the last observation on the attribute was k daysearlier, as x_(i−k,j)+I_(i)−I_(i−k).

In the exemplary embodiment and/or exemplary method, the relationshipbetween the value of the attribute for a particular entity in thepopulation and the conditional index may be defined by a model ofln(x_(ij))=ln(I_(i))+ln(a_(j))+ln(e_(ij)), where x_(ij) is the value ofthe attribute for the jth entity of the population on day i, I_(i), isthe level of the conditional index on day i, a_(j) is a constantassociated with the jth entity of the population, and the e_(ij) haveindependent identically distributed distributions.

The exemplary embodiment and/or exemplary method may further includeestimating the attribute value for at least one entity in the populationwhen the last observation on the attribute value was k days earlier thanday i as x_(i−k,j)(I_(i)/I_(i−k)).

In the exemplary embodiment and/or exemplary method, determining anunconditional index for the attribute may be based on the observationson the attribute and estimates of the value of the attribute. Theunconditional index uses the value of the attribute for a particularentity on a day, conditional only on the entity being part of thepopulation on that day, where the unconditional index represents anaverage level of the attribute for the entities that are in thepopulation at a particular time.

In the above exemplary embodiment and/or exemplary method, thepopulation may include a group of companies having the same creditrating.

In the above exemplary embodiment and/or exemplary method, the attributevalues may include credit derivative pricing data.

In the above exemplary embodiment and/or exemplary method, the attributevalue may include credit default swap spread data for five-year creditdefault swaps.

In the above exemplary embodiment and/or exemplary method, otherless-frequently-observed attributes of the entity of the population maybe determined using a regression analysis where weights declineexponentially back through time. The less-frequently observed attributesmay be non-five-year credit default swap spreads.

The exemplary embodiment and/or exemplary method may further provide agraphical-user-interface to display a data curve for an entity based onthe at least one numerical attribute, where for the entity, the datacurve includes numerical attributes determined using the conditionalindex and the unconditional index.

Another exemplary embodiment and/or exemplary method of the presentinvention provides for determining at least one estimated financialattribute of at least one entity of a population, by: providingfinancial attribute data for entities of the population, wherein thefinancial attribute data is for a period of time, and the period of timeincludes a particular day and prior days; determining a conditionalindex using prior conditional indices, actual financial attribute datafor the particular day and actual financial attribute data for the priordays by calculating a maximum likelihood estimator for the conditionalindex for the particular day; and determining an unconditional indexrepresenting an average level of the financial attribute data for theentities of the population.

Another exemplary embodiment and/or exemplary apparatus of the presentinvention provides for an apparatus for determining an estimate of atleast one numerical attribute of at least one entity of a population ofentities. The population changes and there are a limited number ofobservations on the at least one numerical attribute for the at leastone entity. The apparatus may include a first arrangement to determine aconditional index to track how a value of the at least one numericalattribute changes from one time to another for an entity, which is amember of the population at both times, and a second arrangement todetermine an unconditional index representing an average level of theattribute for the entities that are in the population at a particulartime.

Another exemplary embodiment and/or exemplary apparatus of the presentinvention provides for an apparatus for determining at least oneestimated financial attribute of at least one entity of a population,where the apparatus may include a first arrangement to provide financialattribute data for entities of the population, where the financialattribute: data is for a period of time and the period of time includesa particular day and prior days, and a second arrangement to determine aconditional index for the particular day and for a particular entitybased on a relationship of prior conditional indices, actual financialattribute data for the particular day and actual financial attributedata for the prior days by calculating a maximum likelihood estimatorfor the conditional index for the particular day, and a thirdarrangement to determine an unconditional index representing an averagelevel of the financial attribute data for the entities of thepopulation, and a fourth arrangement to determine the at least oneestimated financial attribute for the at least one entity based on theunconditional index.

Another exemplary embodiment of the present invention provides for acomputer-readable storage medium including program code for determiningan estimate of at least one numerical attribute of at least one entityof a population of entities, where the population changes and there area limited number of observations on the at least one numerical attributefor the at least one entity, in which the program code is executable ina processor arrangement to perform: determining a conditional index totrack how a value of the at least one numerical attribute changes fromone time to another for an entity, the entity is a member of thepopulation at both times; and determining an unconditional indexrepresenting an average level of the attribute for the entities that arein the population at a particular time.

Another exemplary embodiment of the present invention provides for acomputer-readable storage medium including program code for determiningat least one estimated financial attribute of at least one entity of apopulation, where the program code is executable in a processorarrangement to perform: providing financial attribute data for entitiesof the population, where the financial attribute data is for a period oftime and the period of time includes a particular day and prior days;determining a conditional index for the particular day and for aparticular entity based on a relationship of prior conditional indices,actual financial attribute data for the particular day and actualfinancial attribute data for the prior days by calculating a maximumlikelihood estimator for the conditional index for the particular day;determining an unconditional index representing an average level of thefinancial attribute data for the entities of the population; anddetermining the at least one estimated financial attribute for the atleast one entity based on the unconditional index.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an exemplary method of the present invention.

FIG. 2 shows an exemplary system for use with the exemplary method ofFIG. 1 or FIG. 3.

FIG. 3 shows the exemplary method applied for use with CDS spreads.

FIG. 4 shows how the conditional indices for five-year CDS spreadsvaried for the period between Apr. 1, 2001 and May 24, 2002, based ontrading data for five-year CDS spreads.

FIG. 5 shows how the unconditional indices for five-year CDS spreadsvaried for the period between Apr. 1, 2001 and May 24, 2002, based ontrading data for five-year CDS spreads. These are referred to asspread-level indices in the figure.

FIG. 6 shows the parameter a for the period May 31, 2001 to May 24,2002, where the parameter a is the gradient or slope of a credit curvebased on trading data for non-five-year CDS spreads.

FIG. 7 shows the gradient or slope parameter a of the credit curve asmultiplied by the spread-level index to give an estimate of thebasis-point-spread (bps) increase in the CDS spread per year.

FIG. 8A shows a graphical-user-interface of credit calculator software(implemented as an Applet in the Java programming language), which showsa menu structure, a corresponding icon structure for alternatelyimplementing the various menu selections, a “Pricing” section for a listof entities and their financial instrument data, a “Market” section foran entity, and a “Details” section for an entity.

FIG. 8B shows a “File” pull-down menu of the GUI of FIG. 8A.

FIG. 8C shows an “Edit” pull-down menu of the GUI of FIG. 8A.

FIG. 8D shows a “Pricing” pull-down menu of the GUI of FIG. 8A.

FIG. 8E shows a “Format” pull-down menu of the GUI of FIG. 8A.

FIG. 8F shows a “Window” pull-down menu of the GUI of FIG. 8A.

FIG. 8G shows a “Markets-Entities” data section of the “Markets” datasection of the GUI of FIG. 8A.

FIG. 8H shows the “Markets-Credit Curves” data section of the “Markets”section of the GUI of FIG. 8A.

FIG. 8I shows the “Markets-LIBOR” data section of the “Markets” sectionof the GUI of FIG. 8A.

FIG. 8J shows the “Markets-Market Conventions” data section of the GUIof FIG. 8A.

FIG. 8K shows the complete “Details” data section of the “Details” datasection of the GUI of FIG. 8A.

FIG. 8L shows sample “Configure” data for the gridded-square “Report”icon of the GUI (or its corresponding “Report” selection of a pull-downmenu) of FIG. 8A.

FIG. 8M shows a sample “Chart” of the “Report” based on the sample“Configure” data of FIG. 8L.

DETAILED DESCRIPTION

For a certain population of interest there are a number of entities,some of which are members of the population of interest at any giventime. For example, the population of interest may be all entities orcompanies having a particular credit rating or that comprise aparticular industry sector, such as, for example, aerospace, automotive,steel, etc. The set of entities that are members of the population,however, will change through time. In this regard, for example, aparticular company having a credit rating of A might be downgraded toBaa or upgraded to Aa, and therefore leave the population of companieshaving a credit rating of A. Likewise, a company having a credit ratingof Aa may be downgraded to A, and therefore join the population ofcompanies having a credit rating of A.

Furthermore, there will be certain quantifiable attributes of theentities that may be of interest, such as, for example, the price of afinancial instrument. In particular, for example, the attribute ofinterest may be the spread for Credit Default Swap (CDS) instruments,including CDS options (calls or puts) and Cancelable CDS (CCDS)instruments, which are structured using call and put CDS options. For aparticular trading day, there may be an observation of a particularattribute for only some and not all of the entities that are members ofthe population of interest on that day.

In an exemplary apparatus, method and system of the present invention,an index tracking the value of an attribute for all entities that aremembers of the population each day (such as, for example, an index ofCDS spreads for all companies having a credit rating of A) may first bedetermined, and then the attribute(s) for a member of the population areestimated for a day when no observation is available on the attribute(s)for that particular member. In this regard, for example, a spreadattribute for a credit derivative instrument may be estimated for aparticular member of the population of interest. As explained, forexample, the spread attribute may be the spread of a CDS-typeinstrument, or some other credit derivative or other financialinstrument.

While one way to determine the index tracking the value of an attributewould be to average the observations available on the attribute forentities that are members of the population each day, it is believedthat in various circumstances and/or for various conditions, it isbelieved that this approach may produce an index that is too volatilebecause the members of the population for which observations areavailable may differ each day.

FIG. 1 shows an exemplary method of the present invention. The exemplarymethod may be implemented in software using a suitably appropriateprocessor arrangement, which may communicate with other processorarrangements through an Internet/networking arrangement, as in theexemplary system of FIG. 2.

In step 105 of the method 100, the system determines a conditional indexthat describes the average behavior of a particular attribute over aperiod of time (such as a number of days) for a particular entity. Theindex is a conditional index because it is conditional on the entitybeing a member of the population during the period. Thus, for example,the conditional index may describe average changes in the spread of aCDS for a company having a credit rating of A. When the actualobservations for members of the population are not available from anobservational database having the observed attribute data, in step 110,the system estimates attributes for members of the population on thosedays or at those times when actual observations are not available, andthis is performed based on the conditional index. In step 120, thesystem determines an unconditional index by averaging the attributeacross both actual observations of the attribute(s) and estimated“observations” of the attribute(s).

To determine the conditional index, the system uses a model relating thevalue of the attribute for a particular member of the population to theconditional index. Examples of the models that may be used include, forexample, the following ones: (1) x_(ij)=I_(i)+a_(j)+e_(ij); and (2) lnx_(ij)=ln I_(i)+ln a_(j)+ln(e_(ij)) (which may also be expressed as lnx_(ij)=ln(a_(j)I_(I))+ln(e_(ij))), where x_(ij) is the value of theattribute of jth member of the population on day i, I_(i), is the levelof the index on day i, a_(j) is a constant associated with the jthmember of the population, and each of the error terms e_(ij) hasindependent identically distributed distributions. The index I_(i) isarbitrarily set to 100 or some other number for one particular day.Available statistical procedures are then used to determine a maximumlikelihood estimator for the value of the index I_(i) on day i from itsvalue on previous or prior days, the observations of the attribute onday i, and the observations of the attribute on previous days.

In particular, in step 110 of the method 100, the system uses theconditional index to calculate or determine estimated “observations” ofattribute(s) for members of the population when actual observations arenot available. In this regard, assume that it is day i and the attributewas last observed for a particular member of the population k days ago(that is, on day i−k). The system uses the model(s) described above (orsome other suitably appropriate model) to calculate or determine anunbiased estimate of the value of the attribute on day i. For the firstmodel of equation (1), an appropriate estimate would be {circumflex over(x)}_(ij)=x_(i−k,j)+I_(i)−I_(i−k), and for the second model of equation(2), an appropriate estimate would be {circumflex over(x)}_(ij)=x_(i−k,j)I_(i)/I_(i−k), where {circumflex over (x)}_(ij) isthe estimate of the attribute for the jth member of the population onday i.

In step 120 of the method 100, for one or more populations (orcategories), the system calculates or determines the unconditional indexfor each by averaging the particular attribute across both actualobservations and estimated “observations” of the attribute. To calculateor determine the unconditional index on a particular day, the systemconsiders those entities that satisfy the following conditions: (1) theentity is currently a member of the population of interest; and (2) theentity was a member of the population at the time of the most recentobservation on the entity. For these entities, there is either a currentobservation of the attribute or a current estimated value of theattribute. For a population of interest, the system then determines theunconditional index by averaging across the set of current observationsand the current estimated values of the attribute for the entities thatsatisfy both of these conditions.

FIG. 2 shows an exemplary system 200 for implementing or performing theexemplary method 100 of FIG. 1 and the method 300 of FIG. 3, asdescribed herein. The system is configured to execute a programimplementing the exemplary method 100 of FIG. 1. In particular,exemplary processor arrangement 220 executes a program to perform themethod(s) described herein. The processor arrangement 220 may include orbe associated with a computer server arrangement. The program and itsinstructions (which may be an Applet in the Java programming language)for implementing the exemplary methods are accessible and executable bythe processor arrangement 220. The program may be stored in anassociated storage arrangement of the processor arrangement, which maybe a compact disk, hard drive, DVD-ROM, CD-ROM or any other suitablyappropriate and accessible computer-readable storage arrangement and/ormedium, as is well understood. Storage arrangements may be included inor otherwise accessible by the processor arrangement 220, as well as theprocessor arrangements 240 a, 240 b, 240 c, . . . , 240 n of varioussystem user(s). Within the system 200, a system user may use theprocessor arrangement 240 a, 240 b, 240 c, . . . , 240 n to obtain orview the conditional index value, the actual or estimated value(s) ofthe attribute of an entity, and/or the unconditional index that is basedon the conditional index, as well as the trading information at thedatabase 260, which may also be used to store the various index datadescribed herein. The processor arrangements of the system may include apersonal computer, a computer network, a wireless computer or processordevice or arrangement (such as, for example, a PDA) or a wirelesscomputing network, or any other suitably appropriate processorarrangement. The processor arrangements 240 a, 240 b, 240 c, . . . , 240n of the system user(s) may communicate with the processor arrangement220 via a communications network 210, which may be Internet-based.

The processor arrangement 220, uses the program (which may be an Applet,in Java, for example, and which implements the exemplary method) storedat an associated and/or accessible storage arrangement to calculate ordetermine the conditional index, to determine the actual or estimatedattribute value(s), and/or the unconditional index by using theexemplary methods described herein. The database 260 or other storagearrangements store the values of the attribute(s) for the variouspopulation entities (associated with different credit rating or industrysector categories, for example), and may include a database of tradinginformation on various financial instruments, including creditderivatives, such as CDS instruments, which will have various financialterms, including various maturity terms.

The processor arrangement 220 obtains trading data information from andstores such data at the database 260 or other storage arrangements, andmay also store the determined indices and estimated “observation” dataat the database 260 or other storage arrangements. The storagearrangements (including for the database 260) may be accessible via theInternet/communications network 210. The processor arrangement 220 mayuse the Internet/communications network 210 to communicate theconditional indices, the actual or estimated values for the attributesand/or the unconditional indices to the processor arrangements 240 a,240 b, 240 c, . . . , 240 n of the system users.

In a particular example of the exemplary method, the indices of thespreads of CDS instruments may be determined. A trading database havingCDS trading (buy and sell quote) information is referenced to obtain theCDS spread data for determining the conditional and unconditionalindices of CDS spreads for the following exemplary credit ratingscategories of Aaa and Aa, A, and Baa. In particular, the tradingdatabase principally reflects the trading data for five-year CDSinstruments. Both sovereigns and corporations may be included. Theexemplary method is used to determine estimates of the CDS spread forany name and any maturity on any day, based on the trading data for thefive-year CDS spreads. The indices may be calculated or determined oncea day or they may be continuously updated on an intra-day basis.

First, two indices of five-year CDS spreads are determined for eachrating category. The five-year term for CDS spreads is associated withthe majority of trading, so that the use of 5-term CDS spreads onlyreflects the fact that such instruments have been more commonly used.For CDS spreads, the two indices may be referred to as the spread-changeindex and the spread level index, which respectively correspond to theconditional index and unconditional index. The system uses thespread-change index to estimate observations for entities (companies ornames) when actual observations are not available for a particular day.The percentage change in a particular rating category's spread-changeindex between day n₁ and day n₂ is an estimate of the percentage changein the mid-market five-year spread (or some other base term spread)between day n₁ and day n₂ for entities (companies or names) that are inthe rating category on both days. The value of the spread-level indexfor a particular rating category on a particular day represents theaverage mid-market five-year spread for entities (companies or names) inthe rating category on that day.

For outside users of this financial information, only the spread-levelindex may be reported. This enables users to determine whether thespread observed or estimated for a particular entity on a particular dayis high or low relative to the average for the population to which theentity belongs. Thus, a user may obtain this information through theInternet/communications network from the processor arrangement 220, forexample.

In particular, FIG. 3 shows the exemplary method 300 for determining thespread-change index and spread-level index, and for determiningestimates of the various CDS spreads for a particular company (name) oron a particular day (at a particular time, which may be at the end of atrading day). The entities are members of a particular category (such asa credit rating category, or an industry category, for example) at aparticular point in time. The spread-change index and the spread-levelindex may be calculated or determined once a day (such as at the end ofthe day), or they may be continuously determined at any point in a dayif the method is performed on an intra-day basis, rather than at the endof a trading day.

In step 310, the system determines the actual trading observations to beused for a CDS of a particular entity. This may be done in a variety ofways. For example, when there are both bids (offers to buy) and offers(offers to sell) the trading observation may be set equal to the averageof the maximum bid and minimum offer in the last trading day or oversome other recent time period. The determined observation may be storedat the database 260. The bid and offer quotes used to determine anobservation do not have to be simultaneous quotes, so that, for example,the bid quote could be at 10 am and the offer quote could be at 1 pm ona particular day. To better ensure that the bids and offers arereasonably close and not too far apart, the system may require that atleast one of the following two conditions be satisfied:

a<(minoffer−maxbid)/(0.5*(minoffer+maxbid))<b; and

c<(minoffer−maxbid)<d.

where a, b, c, and d are positive or negative constants. The purpose ofthis bid/offer quote restriction is to eliminate or at least reduce dataerrors and situations where the bid and offer quotes are sufficientlyfar apart that the trading data may provide little information about themid-market CDS spread of the company.

Thus, in step 310, the system determines or obtains attribute values orobservations for a plurality of entities, where each entity is a companyor name that is a member of a particular category. In the exemplaryembodiment, the category may be a credit rating category (such as, forexample, Aaa, Aa, A, and Baa), or an industry sector (such as, forexample, automotive, aerospace, biotech, etc.) In step 310, the systemcalculates or determines attribute observations from the attributedatabase. In the exemplary embodiment, the buy/sell quote tradinghistory of a particular entity is for a particular parameter or object,such as the CDS instrument and its spread, and attributes are obtainedfor each of the plurality of entities in the particular category.

In step 320 of the exemplary method, the system determines a conditionalindex that is a spread-change index for all entities of each creditrating category for each day. To provide an initialization orstabilization period, the spread-change index may be started ordetermined some fixed period before the system is fully operational.This initialization or stabilization period may be, for example, about80 business days in the case of five-year CDS spreads. It is believedthat the spread-change index requires an initialization period (such as,for example, 80 days) to provide better predictive results. Thespread-change index I_(i) is the value of the index on day i. The valueof the spread-change index I_(i) is set (arbitrarily) to 100 for each ofthe first N₁ days. The N₁ parameter has a value of 5 days for theparticular CDS example.

To determine the spread-change index I_(i) on a subsequent day, entitiesare identified for which: (1) there is an observation for the entity onthe day; (2) there is an observation for the entity on at least one ofthe previous N₁ days; and (3) the entity is in the particular creditrating category at the time of both observations.

If on day n, there are m entities (companies or names) that satisfy thiscondition and if m is less than 5, then the system sets to I_(n) toI_(n−1). If m is greater than 5, then the following equation may bedefined as follows: u_(j)=ln I_(n−k(j))+ln x_(n,j)−ln x_(n−k(j),j),where x_(ij), is the observation on the jth of the m entities on day iand k(j) is the number of days before day n on which the most recentobservation occurs for this entity. This means that the jth entity hasobservations on day n and on day n−k(j).

In the exemplary method, the parameter Q is defined as the mean of {u₁,u₂, . . . u_(m)}, and the spread-change index for a particular categoryon day n is defined as I_(n)=exp(Q). To reduce the effect of extremeobservations the exemplary method may use the median rather than themean of {u₁, u₂, . . . u_(m)} because it is believed to be moreresistant to extreme or outlying observations than a simple average.

In step 330 of the exemplary method, the system determines theunconditional index or the spread-level index. In particular, the systemdetermines the spread-level index for a particular credit ratingcategory for day n by using the spread-change index for that particularcredit rating category to calculate or determine estimated values of theattribute for all entities (companies or names) on day n that satisfythe following conditions: (1) there is no observation for the entity onday n; (2) there is an observation for the entity on at least one of theN₂ days preceding day n (where N₂ is a parameter whose value isdescribed herein as 30 days for five-year CDS spread data, but which maydiffer (especially based on the particular trading database used)); and(3) the entity is in the credit rating category today and was in therating category at the time of the most recent observation.

Thus, for example, if a particular entity satisfies these conditions andif the most recent observation for the entity is on day n−q, then thesystem determines the estimated observation for that entity on day n asthe actual observation on day n−q multiplied by the ratio ofI_(n)/I_(n−q) (that is, the spread-change index I_(n) on day n dividedby the spread-change index I_(n−q) on day n−q). If there are g actualobservations on day n and if h estimated observations are determined bythe system (in the manner described above), then this provides a totalof (g+h) CDS spreads (or other attribute values for a particularcategory) for day n. The system then determines the spread-level indexfor day n as a central value of these (g+h) CDS spreads (or otherattribute values for a particular category). The central value may bethe mean or, to reduce the impact of extreme observations, the median.

Next, in step 340 of the method, the system determines the attribute(s)for one or more of the particular entities on a particular day. Inparticular, the system determines the attribute (such as, a five-yearCDS spread) as follows. First, if there was an observation on aparticular entity on the prior day (or some other time interval), thenthe system sets the five-year CDS spread equal to that priorobservation. Second, if there was no observation on the prior day, butthere was an observation in the last N₃ days (where N₃ is a parameterwhose value is described herein as 50 days for five-year CDS spreads),then the system sets the estimated five-year CDS spread equal to anestimated observation for the entity. The system determines estimatedobservations, based on the spread-level index as described above, wherethe estimated observation for a particular entity (company or name) on aday n is determined by multiplying the ratio of the spread-change indexI_(n) on day n to the prior spread-change index I_(n−q) on day n−q,(I_(n)/I_(n−q)), by the most recent actual observation on day n−q.Third, if the entity has not traded in the last N₃ days, then the systemsets the five-year CDS spread equal to the current value of thespread-level index for the credit rating category associated with theentity. In step 350, a credit curve is determined. The credit curve maybe based on actual attribute values and estimated attribute values.

In an exemplary use, CDS spread “pricing” software (which may be anApplet in the Java programming language, for example) may be used todetermine the five-year CDS spread for particular companies and this maythen be used to value other credit derivatives. Exemplary userinterfaces for such software are shown in FIGS. 8A to 8L.

FIG. 8A shows a graphical-user-interface of credit curve (data curve)calculator software (implemented as an Applet in the Java programminglanguage for processors 220 and/or 240), which shows a menu structure, acorresponding icon structure for alternately implementing the variousmenu selections, a “Pricing” section for a list of entities and theirfinancial instrument data, a “Market” section for an entity, and a“Details” section for an entity.

With respect to the GUI of FIG. 8A, FIG. 8B shows a “File” pull-downmenu, FIG. 8C shows an “Edit” pull-down menu, FIG. 8D shows a “Pricing”pull-down menu, FIG. 8E shows a “Format” pull-down menu, FIG. 8F shows a“Window” pull-down menu, FIG. 8G shows a “Markets-Entities” data sectionof the “Markets” data section, FIG. 8H shows the “Markets-Credit Curves”data section of the “Markets” section, FIG. 8I shows the “Markets-LIBOR”data section of the “Markets” section, FIG. 8J shows the “Markets-MarketConventions” data section, FIG. 8K shows the complete “Details” datasection of the “Details” data section, FIG. 8L shows sample “Configure”data for the gridded-square “Report” icon of the GUI (or itscorresponding “Report” selection of a pull-down menu) of FIG. 8A, andFIG. 8M shows a sample “Chart” of the “Report” based on the sample“Configure” data of FIG. 8L.

In particular, the credit curve of FIGS. 8A and 8H shows the spread“pricing” (or other numerical attribute values for a particularcategory) structure for various terms for a financial instrument, suchas a CDS, for a selected entity on a particular trading day. Thus, forexample, for a CDS spread credit curve (based on five-year CDS spreaddata from a CDS buy/sell trading information database), CDS spread“pricing” may be provided for various maturity terms (such as, forexample, six months, one year, two years, three years, four years, fiveyears, seven years and ten years), shown in FIGS. 8A and 8H. The creditcurve has a gradient or slope parameter a (which is described furtherbelow), and default probabilities may be determined from the base creditcurve.

These default probabilities associated with the CDS spreads may bedetermined using the valuation and default probability methodologydescribed in the papers “Valuing Credit Default Swaps I: No CounterpartyDefault Risk” by John Hull and Alan White, published in the Journal ofDerivatives, Vol. 8, No. 1, (Fall 2000), pp. 29-40 vol., which isincorporated herein by reference in its entirety and in particular as tothe default probability methodology described in that paper, and by“Valuing Credit Default Swaps II: Modeling Default Correlations” by JohnHull and Alan White, published in the Journal of Derivatives, Vol. 8,No. 3, (Spring 2001), pp. 12-22, which is also incorporated herein byreference in its entirety and in particular as to the defaultprobability methodology described in that paper.

In the CDS spread (or other financial instrument or credit derivative)pricing software of FIG. 8A, the software may have information oncertain entities but not others. The software may provide a list of thetracked entities to a user through the graphical user interfaces (GUIs)of FIGS. 8A to 8M. If the user enters an entity name that is not on theentity list, the user must supply the entity credit curve information,which may be based on the credit curve information for a similar entity.Thus, for example, if Daimler-Chrysler Co. is not on the list, it may bepriced based on the spread credit curve information for Ford MotorCompany.

As described above, the trading information database 260, for example,may include trading statistics primarily for five-year CDS spreads, sothat the system estimates the five-year CDS spreads for one or moreentities for each credit rating category on a particular day, for whichthere are no actual observations. For example, if 85% of the tradingdata in the database reflects bid and offer quotes for five-year CDSspreads, then the indexes described herein would be for five-year CDSinstruments.

To complete the credit curve beyond the base-term maturity of five-yearCDS spreads, however, the system in step 340 also determines estimatesof non-five-year spreads for each of the entities for each credit ratingcategory on a particular day. Accordingly, when there are bid and offerquotes for a CDS instrument having a maturity term that is differentthan five years, a CDS spread observation for a maturity term other thanfive years may be calculated or determined in the same manner in whichthe system determines five-year spread observations. This includesapplying the same criteria described above to ensure that the maxbid andminoffer are sufficiently close and therefore not too far apart.

A model for relating non-five-year spreads to five-year spreads may beas follows: Spread(k)=Spread(5)+a*Spread(5)*(k−5), where Spread(k) isthe spread for a k-year CDS instrument having a k-year term. Once theparameter a (which is the slope or gradient of the credit curve) hasbeen estimated, the system may determine estimates for all non-five-yearCDS spreads from the estimates of five-year CDS spreads. More generally,the model may be expressed as follows:Spread(k)=Spread(BM)+a*Spread(BM)*(k−BM), where BM is the base maturityterm, which in the specific example is 5 years. This model is used toprovide the credit curve data for non-five-year maturities in FIGS. 8Aand 8H, where the gradient or slope parameter a is displayed.

The system determines the slope or gradient parameter a of the creditcurve by using a regression analysis in which the weights assigned tothe observations are decreased exponentially as the system looks orsearches for observations further back in time. To provide aninitialization or stabilization period, the exemplary method begins byobserving non-five-year data M days before the start date for thespread-level index. A value of 80 days for M is believed to work wellwith a database largely based on five-year CDS spread trading data, butM may vary depending on the nature of the particular observation data.The first M days of observations constitute the initialization orstabilization period.

For each non-five-year observation on each day, the system firstattempts to determine an estimate of the corresponding five-year spreadusing the first and second steps described above for estimatingfive-year spreads for entities. If the corresponding five-year spread isnot determinable or if the five-year spread is more than three times thelevel of the spread-level index, then the system discards the data. Thesystem may discard the data when the estimated five-year spread is morethan three times the spread-level index to eliminate extremeobservations that may not be representative of the particular ratingcategory. An example of an “extreme” observation from the A-ratedcategory is an entity such as, for example, WorldCom, which in the firstfew months of 2002, had a five-year spread of about 600 and a 1-yearspread of about 800. In contrast, it is believed that many or mostA-rated companies had a five-year spread that was greater than—and notless than—the 1-year spread. Accordingly, it is believed that it may beinappropriate or less predictive to use such “extreme” data to estimateanything about the term structure of credit spreads for “typical”companies having an A credit rating.

If after the data filtering described above, there are, for example,m_(i) non-five-year spread observations on day I, in the exemplarymethod as applied to CDS instruments, the following parameters may bedefined as follows: U_(ij) as the spread for the non-five-year CDS forthe jth observation on day i; V_(ij) as the spread for the correspondingfive-year CDS for the jth observation on day i; T_(ij) as the life ofnon-five-year CDS for the jth observation on day i; X_(ij)

${{as}\mspace{14mu} {V_{ij}\left( {T_{i,j} - 5} \right)}};{{y_{ij}\mspace{14mu} {as}\mspace{14mu} U_{ij}} - V_{ij}};{\alpha_{i} = {\sum\limits_{j = 1}^{m_{i}}{x_{i,j}y_{i,j}}}};{{{and}\mspace{14mu} \beta_{i}} = {\sum\limits_{j = 1}^{m_{i}}{x_{i,j}^{2}.}}}$

The value of the slope or gradient parameter a of the credit curve onday i is given by A_(i)/B_(i), where A_(i) and B_(i) are parameters thatmay be updated by the system each day. They are first calculated on dayM from the M days of the initialization or stabilization period. On dayM, the parameter w_(M)=1−λ, where λ is a parameter used to provide theeffect of exponentially decreasing the weights assigned to theobservations, and whose value is described below as 0.99 based onfive-year CDS trading data. For values of i from 1 day to M−1 days, thesystem uses w_(i)=λw_(i+1). The initial values of A_(M) and B_(M) are asfollows:

$A_{M} = \frac{\sum\limits_{i = 1}^{M}{w_{i}\alpha_{i}}}{\sum\limits_{i = 1}^{M}w_{i}}$$B_{M} = \frac{\sum\limits_{i = 1}^{M}{w_{i}\beta_{i}}}{\sum\limits_{i = 1}^{M}w_{i}}$

For i greater than M days, the system determines A_(i) and B_(i) asfollows: A_(i)=λA_(i−1)+(1−λ)α_(i); and B_(i)=λB_(i−1)+1−λ)β_(i).

As described above, the exemplary method and system determines theconditional index (the spread-change index for CDS spreads) and theunconditional index (the spread-level index for CDS spreads) and CDSspread estimates by using four parameters: N₁, N₂, N₃, and λ. Based onthe five-year CDS spread data of an exemplary CDS/trading database, inwhich about 85% of the data was for five-year CDS instruments, theexemplary values for these parameters may be N₁=5 days, N₂=30 days,N₃=50 days, and λ=0.99, where λ is used to exponentially decrease theobservation data as the data becomes older, as described herein.

For different credit rating categories and for the five-year CDS spreaddata, FIG. 4 shows the conditional indices (e.g., spread-change indices)and how they varied between Apr. 1, 2001 and May 24, 2002, FIG. 5 showsthe unconditional indices (e.g., spread-level indices) between Apr. 1,2001 and May 24, 2002, FIG. 6 shows the gradient or slope parameter a ofthe credit curve based on trading data for non-five-year CDS spreads forthe period May 31, 2001 to May 24, 2002, and FIG. 7 shows the gradientor slope parameter a of the credit curve as multiplied by thespread-level index to provide an estimate of the basis-point-spread(bps) increase in the CDS spread per year.

Thus, the exemplary method may be used to estimate five-year CDSspreads. For the estimation, it is assumed that the CDS spread for aparticular company behaves in the same way as CDS spreads for othersimilar companies. To estimate the CDS spread for a particular A-ratedcompany on November 15, for example, the following may be assumed: (1)the most recent, reliable information available on the company was onNovember 8 when the maximum bid and minimum offer for the particularcompany were 100 and 120 basis points, respectively; and (2) spreads forA-rated names have increased by 5% on average between November 8 andNovember 15. The estimate for the spread on November 15 would then bethe mid-market spread on November 8 grossed up by 5%. That is, it wouldbe 110×1.05=115.5.

The estimate of a non-five year CDS spread is based on the slopes of thelines that relate CDS spread to CDS life for different ratingcategories. If, for example, on November 15, it is estimated that CDSspreads for A-rated companies increase by 3% for each year of the lifeof the CDS, then the 7-year CDS spread for the company in the examplewould be 115.5+2×0.03×115.5=122.43. The estimates assume that the CDSspread for the company under consideration moves similarly to the CDSspreads for other companies with similar credit ratings. If there havebeen no recent updates about the creditworthiness of the company, theestimate may be very good. If new updates have recently been releasedcausing the market to revise its opinion about the company, the estimatemay be less favorable.

To test the procedure for estimating CDS spreads, the following inquirymay be made for each company on each day for which there were bids andoffers that were reasonably close together: “If the bids and offers hadnot been observed, what would have been the estimated CDS spread?” The“error” may then be determined as the absolute difference in basispoints between the estimate and the mid-point between the maximum bidand minimum offer. Results for the period April 2001 to May 2002 aresummarized in Table 1. The calculated error may overstate the actualerror as the market CDS spread could be anywhere between the maximum bidand the minimum offer.

TABLE 1 Aaa/Aa A Baa Median Error (bps) 1.16 2.01 3.82 Upper QuartileError (bps) 2.90 4.48 8.97 Average of Difference 7.45 11.13 20.12Between MinOffer and MaxBid (bps) Number of Estimates 2,659 9,585 8,170Tested

In Table 1, it is shown that 2,659 tests were performed for Aaa/Aacompanies. In 50% of the cases, errors were less than 1.16 basis pointsand in 75% of the cases errors were less than 2.90 basis points. Theaverage spread between the minimum offer and maximum bid observed was7.45 basis points. This means that the market CDS spread could bedifferent from what is assumed to be the true CDS spread by as much as3.725 basis points on average. The results for other rating categoriesmay be interpreted similarly. Available non-five-year CDS trading datafor testing the accuracy of the estimates on non-five-year spreadsindicates that the errors should be comparable or similar to those forfive-year spreads.

As explained above, the trading information database 260 is updated eachday to include determined bid (buy) quote and offer (sell) quote trading“observations.” In particular, when there are both bid and offer quotesfor a CDS for a particular name on a particular day and the bid-offerspread is sufficiently small (according to criteria described herein),the system determines an observation for that name and that day as0.5*(maxbid+minoffer), where maxbid is the maximum of the bids on theday and minoffer is the minimum of the offers on the day.

The spread-change index is calculated using the model ofln(x_(ij))=ln(a_(j)I_(i))+ln(e_(ij)), where I_(i) is the index on day i,x_(ij) is the five-year spread for the jth company on day j, a_(i)i,a_(j) is a constant for company j, and the e_(ij) are independent,identically distributed variables. In this equation the five-year spreadfor a particular entity (company or name) has a component that dependson the index level and a component that is unique to the entity (companyor name), and the system uses available maximum likelihood estimatorstatistical techniques to estimate the spread-change index I_(i). Thespread-change index for Apr. 1, 2001 to May 24, 2002 is shown in FIG. 4,in which the indices for all rating categories were set equal to 100 onJan. 1, 2001.

The system determines the spread-level index for a rating category on aparticular day as the average of a set of five-year spreads. The setconsists of a five-year spread observations for names in that categoryon that day, and five-year spread estimates for other names on that daycalculated using the spread-change index. The objective is to ensurethat the set of names over which the average is determined remainsreasonably stable through time. The spread-level index for Apr. 1, 2001to May 24, 2002 is shown in FIG. 5.

Also, as described herein, the five-year spread for a name on a day isestimated as follows: (1) if there is an observation for the five-yearspread for the name on the day, then the 5-year spread is set equal tothat observation; (2) if there is no observation for the five-yearspread for the name on the day, but there are recent such observations,then the five-year spread is estimated from the recent observationsusing the spread-change index; and (3) if there are no recentobservations for the five-year spread for the name, then the five-yearspread is set equal to the spread-level index.

In the exemplary method and system described above, the spread-changeindex and the spread-level index are determined or updated once a day.They may also be updated on a continual basis or intra-day basis. Inparticular, the method and system may be used to determine thespread-change index and the spread-level index at a particular timeduring the day in the same manner as if it were the end of the tradingday. This means that an index for a particular rating category may beupdated when there are new observations on a sufficient number ofentities (such as, for example, five entities) in the credit ratingcategory, and when for each of these entities there is also at least oneobservation in the last N₁ days.

In particular, the above-described methods and system may be used todetermine estimates of five-year CDS spreads on particular entities, andthese may be updated on an intra-day basis. As described above, usingthe methods described herein, the system determines the value of anobservation for an entity on a particular day, by setting theobservation equal to 0.5*(maxbid+minoffer), where maxbid is the maximumof all bid quotes during the day and minoffer is the minimum of alloffer quotes during the day. To use this method when there is intra-dayupdating of the observation, the system determines maxbid and minofferfrom all bid and offer quotes up to a particular time on that particularday, where recent bid and offer quotes would be the most relevant.

To better understand the spread-change index and the spread-level indexdescribed herein, it should be understood that they may behavedifferently, as evidenced by the following analogy. To construct anannual index of the ages of people in the United States, the increase inthe index each year may be set equal to the average increase in the ageof people who are alive at both the beginning of the year and at the endof the year, or the index each year may be set equal to the average ageof the population. The first index grows at the rate of 1 year per year.The second, index may be growing, but much more slowly.

The spread-change index of CDS spreads is analogous to the first indexof age, and the spread-level index is analogous to the second index ofage. Consider the A rating category, where companies in the categoryappear on average to be getting less credit-worthy with the passage oftime. As a result, the average change in the CDS spread for a group ofA-rated companies may increase so long as the companies continue to havean A credit rating. This increase is measured by the spread-changeindex. When the credit-worthiness of one of the companies declines tobelow a certain level, the company is downgraded and therefore leavesthe A-rated sample. Also, there are periodic infusions of new A-ratedcompanies because of downgrades from higher ratings. This is analogousto births and deaths in the above aging example. As a result, thepercentage change in the average CDS spread for A-rated companies (whichis measured by the spread-level index) may tend to be not as great asthe percentage change in the spread-change index.

1-14. (canceled)
 15. A method for determining at least one estimatedfinancial attribute of at least one entity of a population, the methodcomprising: providing financial attribute data for entities of thepopulation, wherein the financial attribute data is for a period oftime, and the period of time includes a particular day and prior days;determining a conditional index for the particular day and for aparticular entity based on a relationship of prior conditional indices,actual financial attribute data for the particular day and actualfinancial attribute data for the prior days by calculating a maximumlikelihood estimator for the conditional index for the particular day;determining an unconditional index representing an average level of thefinancial attribute data for the entities of the population; anddetermining the at least one estimated financial attribute for the atleast one entity based on the unconditional index.
 16. (canceled)
 17. Anapparatus for determining at least one estimated financial attribute ofat least one entity of a population, the apparatus comprising: a firstarrangement to provide financial attribute data for entities of thepopulation, wherein the financial attribute data is for a period oftime, and the period of time includes a particular day and prior days; asecond arrangement to determine a conditional index for the particularday and for a particular entity based on a relationship of priorconditional indices, actual financial attribute data for the particularday and actual financial attribute data for the prior days bycalculating a maximum likelihood estimator for the conditional index forthe particular day; a third arrangement to determine an unconditionalindex representing an average level of the financial attribute data forthe entities of the population; and a fourth arrangement to determinethe at least one estimated financial attribute for the at least oneentity based on the unconditional index.
 18. (canceled)
 19. Acomputer-readable storage medium including program code for determiningat least one estimated financial attribute of at least one entity of apopulation, the program code being executable in a processor arrangementto perform the following: providing financial attribute data forentities of the population, wherein the financial attribute data is fora period of time, and the period of time includes a particular day andprior days; determining a conditional index for the particular day andfor a particular entity based on a relationship of prior conditionalindices, actual financial attribute data for the particular day andactual financial attribute data for the prior days by calculating amaximum likelihood estimator for the conditional index for theparticular day; determining an unconditional index representing anaverage level of the financial attribute data for the entities of thepopulation; and determining the at least one estimated financialattribute for the at least one entity based on the unconditional index.